Expanding the Lexicon: Analyzing Nouns, Adjectives, and Transitive Verbs, Study notes of Linguistics

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Expanding the Lexicon

LING 255

January 31, 2008

1. BEYOND VERBS

Last time we ended with:

(1) The Rule of “Noun Verb” interpretation If we have a Sentence node, and its daughters are NP and VP, then

S

[[NP VP]] = TRUE if [VP]

[[snores]] = the characteristic function of { x | x snores}, i.e., that function from individuals to truth values such that, given an individual, it returns TRUE if and only if that individual snores.

That may cover verbs...what about nouns?

1.1. Nouns

• [[Lincoln is a president]] = TRUE iff Lincoln is a president. [[president]] = ...?

We have two options: one is to assign meanings to is, a, and president; the other is to ignore the little words and assume that the meaning comes only from Lincoln and president. Let’s ignore is and a. Then, since all we have is function application, we’ll end up with a meaning much like the one we had above:

[[president]] = that function from individuals to truth values such that, given an individual, it returns TRUE if and only if that individual is a president.

1.2. Adjectives

Do adjectives work in exactly the same way?

• [[Lincoln is happy]] = TRUE iff Lincoln is happy. [[happy]] = that function from individuals to truth values such that, given an individual, it returns TRUE if and only if that individual is happy.

Apparently so. [But stay tuned!]

Instructor: Lance Nathan

1.3. Some concerns about these meanings

• Can we really justify treating is and a as meaningless?

Well, consider Hebrew: the sentence Lincoln is a president is expressed as Lincoln nasi,^1 with no word for is or a. (Adjectives are the same: there’s no is in the Hebrew translation of Lincoln is happy.)

Our two options can be recast as follows: either we combine “Lincoln” directly with “president”, and English has a few words without meaning; or we include “is” and “a”, and Hebrew (along with, e.g., Chinese) expresses these words invisibly. The first option seems much simpler...

• Aren’t nouns more like individuals?

Possibly. But a noun like “president” doesn’t just name a particular individual. Names of individuals can be swapped around freely, e.g.

Hesperus, the evening star, is Venus. Bosperus, the morning star, is Venus. Therefore, Hesperus is Bosperus.

That doesn’t work so well with:

Abraham Lincoln is X, the individual denoted by “president”. George W. Bush is X, the individual denoted by “president”. Therefore, Abraham Lincoln is George W. Bush.

So we need something more complex than just an individual. (It is the case that nouns get used to pick out individuals, e.g. the dog barked or the president snores; but not on their own! The word the has to be doing some work there—similarly in, say, Hebrew, where ha-nasi picks out the individual.)

• If [[is a president]] = that function from individuals to truth values such that, given an individual, it returns TRUE if and only if that individual is a president, then will we have:

[[is the president]] = that function from individuals to truth values such that, given an individual, it returns TRUE if and only if that individual is the president?

Well...by hypothesis, a isn’t contributing any meaning. On the other hand, the presumably is (as was just claimed above). We don’t want to just freely substitute α for β in the object language and then substitute α for β in the metalanguage.

For instance...

(^1) I had “naasi” on the board in class, but I gather “nasi” is the usual representation.

Instructor: Lance Nathan

2. BACK TO VERBS: TRANSITIVE VERBS

• [[Booth killed Lincoln]] = TRUE iff Booth shot Lincoln

How do we analyze this? We don’t want to just say:

• [[killedlincoln]] = that function from individuals to truth values such that, given an individual, it returns TRUE if and only if that individual killed Lincoln

(and then [[killedkennedy]] is the same if we substitute “Kennedy” for “Lincoln” in the metalanguage description, and so forth)

...because here, we want killed and Lincoln to each contribute their separate meanings.

The good news: that’s not too hard to do, as long as our functions can take ordered pairs as arguments.

[[killed]] = that function from pairs of individuals to truth values such that, given an pair of individuals < x , y >, it returns TRUE if and only if x killed y

The bad news is that we need a new interpretation rule, because Booth killed Lincoln won’t just have the S-over-NP-and-VP structure, but instead something like:

(2) S

NP V NP | | | Booth killed Lincoln

So we’ll need a new rule like the following:

(3) The Rule of Transitive Verb Interpretation If we have a Sentence node, and its daughters are NP, V, and NP, then

S

[[NP 1 V NP 2 ]] = TRUE if [V]

That’s not too bad, though it’s not great...but worse, we now have to have a different rule for Japanese, where the word order is Subject-Object-Verb:

Instructor: Lance Nathan

(4) The Rule of Transitive Verb Interpretation in Japanese If we have a Sentence node, and its daughters are NP, NP, and V, then

S

[[NP 1 NP 2 V]] = TRUE if [V]

(Why not just say “If we have a sentence node with these three daughters, in whatever order?” Because we have to be able to distinguish NP 1 from NP 2 .)

And there are other reasons to think that perhaps shot Lincoln is a unit of the sentence, whereas Booth shot is not. What we need is a vast simplification of things.

3. THE RULE OF INTERPRETATION, REVISITED

We started with a very particular rule of interpretation for S = NP VP. Instead, we can use a much more general rule:

(5) Function Application (Rule #1 of Interpretation) If a node M has daughters D 1 , D 2 [in either order!], and D 1 is a function that can take D 2

as an argument, then [[M]] = [[D 1 ]]([[D 2 ]])

You can verify for yourself that the rule in (1) is just a special case of this rule.

Now the only problem is that we can only combine two things at a time, not three, and our semantics for killed is such that it requires an ordered pair, rather than a single individual.

Fortunately, this can be done with...Schönfinkelization!